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Adaptivity and symmetry for ODEs and PDEs Chris Budd

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Talk will look at variable step size adaptive methods for ODES scale invariant adaptive methods for PDES Basic Philosophy ….. ODES and PDEs develop structures on many time and length scales Structures may be uncoupled (eg. Gravity waves and slow weather evolution) and need multi-scale methods Or they may be coupled, typically through (scaling) symmetries and can be resolved using adaptive methods

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Conserved quantities : Symmetries: Rotation, Reflexion, Time reversal, Scaling Kepler's Third Law HamiltonianAngular Momentum The need for adaptivity: the Kepler problem

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Kepler orbits Stormer Verlet Symplectic Euler Forward Euler

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Global error H error FE SV t Main error Larger error at close approaches Keplers third law is not respected

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Adaptive time steps are highly desirable for accuracy and symmetry But … Adaptivity can destroy the symplectic shadowing structure [Calvo+Sanz-Serna] Adaptive methods may not be efficient as a splitting method AIM: To construct efficient, adaptive, symplectic methods EASY which respect symmetries

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H error t

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The Sundman transform introduces a continuous adaptive time step. IDEA: Introduce a fictive computational time Hamiltonian ODE system : SMALL if solution requires small time-steps

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Can make Hamiltonian via the Poincare Transform New variables Hamiltonian Rescaled system for p,q and t Now solve using a Symplectric ODE solver

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Choice of the scaling function g(q ) Performance of the method is highly dependent on the choice of the scaling function g. Approach: insist that the performance of the numerical method when using the computational variable should be independent of the scale of the solution and that the method should respect the symmetries of the ODE

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The differential equation system Is invariant under scaling if it is unchanged by the symmetry It generically admits particular self-similar solutions satisfying eg. Keplers third law relating planetary orbits

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Theorem [B, Leimkuhler,Piggott] If the scaling function satisfies the functional equation Then Two different solutions of the original ODE mapped onto each other by the scaling transformation are the same solution of the rescaled system scale invariant A discretisation of the rescaled system admits a discrete self-similar solution which uniformly approximates the true self-similar solution for all time

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Example: Kepler problem in radial coordinates A planet moving with angular momentum with radial coordinate r = q and with dr/dt = p satisfies a Hamiltonian ODE with Hamiltonian If symmetry Numerical scheme is scale-invariant if

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If there are periodic solutions with close approaches Hard to integrate with a non-adaptive scheme q t

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Consider calculating them using the scaling No scaling Levi-Civita scaling Scale-invariant Constant angle change

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H Error Method order Surprisingly sharp!!!

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Scale invariant methods for PDES These methods extend naturally to PDES with scaling and other symmetries

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Examples Parabolic blow-up High-order blow-up NLS Chemotaxis PME Rainfall Need to continuously adapt in time and space Introduce spatial analogue of the fictive time

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Adapt spatially by mapping a uniform mesh from a computational domain into a physical domain Use a strategy for computing the mesh which takes symmetries into account

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Introduce a mesh potential Geometric scaling Control scaling via a measure

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Spatial smoothing (Invert operator using a spectral method) Averaged measure Ensures right-hand- side scales like P in d-dimensions to give global existence Parabolic Monge-Ampere equation PMA (PMA) Evolve mesh by solving a MK based PDE

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Because PMA is based on a geometric approach, it has natural symmetries 1. System is invariant under translations and rotations 2. For appropriate choices of M the system is invariant under scaling symmetries

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PMA is scale invariant provided that

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Example: Parabolic blow-up in d dimensions Scale: Regularise:

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Basic approach Discretise PDE and PMA in the computational domain Solve the coupled mesh and PDE system either (i) As one large system (stiff!) or (ii) By alternating between PDE and mesh Method admits exact discrete self-similar solutions

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solve PMA simultaneously with the PDE Mesh: Solution: X Y 10 10^5

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Solution in the computational domain 10^5 Same approach works well for the Chemotaxis eqns, Nonlinear Schrodinger eqn, Higher order PDEs Now extending it to CFD problems: Eady, Bousinessq

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